Dynamics of droplets in an agitated dispersion with multiple breakage. Part I: formulation of the model and physical consistency Antonio Fasano, Fabio Rosso Mathematics Subject Classification (1991): 82C99,76T20,76T99,45K05 This work was partially supported by the G.N.F.M.–I.N.D.A.M. Strategic Project “Metodi Matematici in Fluidodinamica e Dinamica Molecolare” Abstract. In [7] a new model for the evolution of a system of droplets dispersed in an agitated liquid was presented, with the inclusion of the so–called volume scattering effect (a combination of coalescence and breakage). In that paper droplets breakage was considered to be binary, in order to simplify exposition. Here we remove that limitation, investigating the effect of each breakage mode and of scattering with multiple exits. We also allow the breakage kernel, at each mode, to become singular when droplets approach their finite maximum admissible size. 1. Introduction A system of two immiscible liquids agitated in a batch under the action of impellers gives rise to a set of droplets of one phase dispersed in the other phase. The resulting system is called a dispersion (finer dispersions are called emulsions) and its evolution is caused by the fact that droplets during their motion may break up in two or more smaller droplets or they may coalesce (an essentially binary process), producing larger elements. Dispersions are commonly encountered food industry, cosmetics, pharmacology, photography and many others industrial processes . This justifies the large amount of scientific papers devoted to this subject during the last century (see, for example, [1,2,4] for the main relevant literature). However, many basic questions are still pending so that research is still very active in this area (see, for example, [5,8,12,14,15,10]). Dealing with the specific case of the batch reactor, it is commonly assumed that spatial ho- mogeneity is achieved, so that the droplet system is described by a volume distribution func- tion f , so that f (v,t)dv represents the number of droplets having volume in the interval